Preheat the oven: Preheat the oven to 425°F and line a large baking sheet with parchment paper. A hot oven from the start helps the sweet potatoes crisp instead of steam.
Cube the sweet potatoes: Peel the sweet potatoes and cut them into even 3/4-inch cubes so every piece cooks at the same rate. Pat the cubes dry with a paper towel to remove excess moisture.
Season the potatoes: In a large bowl, toss the sweet potato cubes with olive oil, salt, pepper, and smoked paprika until evenly coated. Make sure the oil reaches every side of each cube.
Arrange in a single layer: Spread the seasoned cubes on the prepared baking sheet in one even layer, leaving a little space between pieces. Crowding the pan causes steaming, so use a second pan if needed.
Roast the first batch: Roast for about 20 minutes, until the bottoms of the cubes start turning golden brown. Avoid stirring during this time so the undersides get a chance to caramelize.
Flip and add garlic: Remove the pan, flip the sweet potato cubes, and scatter the minced garlic evenly over the top. Adding garlic now keeps it from burning over a longer roast.
Finish roasting: Return the pan to the oven for another 10 to 12 minutes, until the edges are deeply caramelized and a fork pierces the cubes with no resistance. The garlic should look soft, not burnt.
Add the parmesan: Pull the pan from the oven and immediately sprinkle the parmesan cheese over the hot potatoes so it begins to melt. Toss gently to distribute the cheese.
Melt the cheese: Slide the pan back into the oven for 2 to 3 minutes, just until the parmesan turns golden in spots. Watch closely, as cheese can go from melted to burnt quickly.
Garnish and serve: Scatter the chopped fresh parsley over the top for a burst of color and a fresh, grassy note. Transfer to a serving dish while warm.